We examined children's development of strategic and conceptual knowledge for linear measurement. We conducted teaching experiments with eight students in grades 2 and 3, based on our hypothetical learning trajectory for length to check its coherence and to strengthen the domain-specific model for learning and teaching. We checked the hierarchical structure of the trajectory by generating formative instructional task loops with each student and examining the consistency between our predictions and students' ways of reasoning. We found that attending to intervals as countable units was not an adequate instructional support for progress into the Consistent Length Measurer level; rather, students must integrate spaces, hash marks, and number labels on rulers all at once. The findings have implications for teaching measure-related topics, delineating a typical developmental transition from inconsistent to consistent counting strategies for length measuring. We present the revised trajectory and recommend steps to extend and validate the trajectory. Measurement of geometric space is an important aspect of the broad curriculum of mathematics in elementary education, yet research on teaching and learning measurement is less robust and less extensive than research on other domains such as number. Further, the research on learning measurement has not been thoroughly integrated with instructional design and pedagogical concerns. The goal of this study is to implement and improve a learning trajectory in the domain of measurement. We are particularly interested in the potential integration of developmental progressions that sequence increasingly sophisticated types of thinking in a domain with pedagogical progressions that list appropriate types of tasks and instructional activities suited to specific levels along that developmental progression. By integrating learning progressions and pedagogical progressions within the context of theoretical accounts of children's cognitive operations on mental images and concepts, we expect to forge more practical and productive learning trajectories. Such trajectories provide important infrastructure for improving assessment tools, professional development models, and curriculum design.
A 14-year-old child with Acute Lymphoblastic Leukemia participated in 52 weeks of robotics task-based interviews. We present 3 of her tasks from Weeks 1, 20, and 46 along with an overview of the complete 52 weeks. We compare the data from the tasks to Brousseau's (1997) Theory of Didactical Situations of Mathematics to answer our research questions: Can robotics play support the devolution of a fundamental situation to an adidactic situation of mathematics for children who are critically ill? When children with critical illness engage in robotics play, what are the key features of the robotics phenomenon that support devolution to an adidactic situation? We found evidence of the robotics supporting the devolution of a fundamental situation to an adidactic situation of mathematics in each robotics task and evidence of 4 key features (thick authenticity, feedback enabling autonomy, connectivity, and competence) of robotics play that support this devolution.
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